Weighted partial sums of a random multiplicative function and their positivity

In this paper, we study the probability that some weighted partial sums of a random multiplicative function f are positive. Applying the characteristic decomposition, we obtain that if S is a non-empty subset of the multiplicative residue class group (Z/mZ)× with m being a fixed positive integer and A=+mn n=0,1,2,3,⋯\ with a∈ S, then there exists a positive number $δ$ independent of x, such that \[ P(∑A∩[1,x)f(n)/n<0)>δ\] unless the coefficients of the real characters in the expansion of the characteristic function of S according to the characters of (Z/mZ)× are all non-negative, and the coefficients of the complex characters are all zero, in which case we have \[ P(∑A∩[1,x)f(n)/n<0)=O(exp(-exp({ln x}{Cln₂x}))) \] for a positive constant $C.$ This includes as a special case a result of Angelo and Xu. We also extend the result to the cyclotomic field Kₙ=Q(ζₙ) with ζₙ=e2πi/n and study the probability that these generalized weighted sums are positive. In addition, we deal with the positivity problem of certain partial sums related to the celebrated Ramanujan tau function $τ(n)$ and the Ramanujan modular form $Δ(q),$ and obtain an upper bound for the probability that these partial sums are negative in a more general situation.